# Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves)

### Josep González Rovira

Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain### Joan-Carles Lario

Universitat Politècnica de Catalunya, Barcelona, Spain

## Abstract

Let $A_{f}$ be the abelian variety attached by Shimura to a normalized newform $f∈S_{2}(Γ_{1}(N))$ and assume that $A_{f}$ has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in $S_{2}(Γ_{1}(N))$ corresponding to the pullbacks of the regular differentials of all elliptic quotients of $A_{f}$. For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in [8]. The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then we apply the results obtained to the case $N=p_{2}$, for primes $p>3$ and $p≡3$ mod $4$. For this case we prove that if $f$ has CM, then all optimal elliptic quotients of $A_{f}$ are also optimal in the sense that its endomorphism ring is the maximal order of $Q(−p )$. Moreover, if $f$ has trivial Nebentypus then all optimal quotients are Gross’s elliptic curve $A(p)$ and its Galois conjugates. Among all modular parametrizations $J_{0}(p_{2})→A(p)$, we describe a canonical one and discuss some of its properties.

## Cite this article

Josep González Rovira, Joan-Carles Lario, Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves). Comment. Math. Helv. 86 (2011), no. 2, pp. 317–351

DOI 10.4171/CMH/225